Algebra Examples

$\frac{a}{a - b} + \frac{3 a}{a + b} - \frac{2 a b}{a^{2} - b^{2}}$

Step 1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = a$ and $b = b$ .

$\frac{a}{a - b} + \frac{3 a}{a + b} - \frac{2 a b}{(a + b) (a - b)}$

Step 2

To write $\frac{a}{a - b}$ as a fraction with a common denominator, multiply by $\frac{a + b}{a + b}$ .

$\frac{a}{a - b} \cdot \frac{a + b}{a + b} + \frac{3 a}{a + b} - \frac{2 a b}{(a + b) (a - b)}$

Step 3

To write $\frac{3 a}{a + b}$ as a fraction with a common denominator, multiply by $\frac{a - b}{a - b}$ .

$\frac{a}{a - b} \cdot \frac{a + b}{a + b} + \frac{3 a}{a + b} \cdot \frac{a - b}{a - b} - \frac{2 a b}{(a + b) (a - b)}$

Step 4

Write each expression with a common denominator of $(a - b) (a + b)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{a}{a - b}$ by $\frac{a + b}{a + b}$ .

$\frac{a (a + b)}{(a - b) (a + b)} + \frac{3 a}{a + b} \cdot \frac{a - b}{a - b} - \frac{2 a b}{(a + b) (a - b)}$

Step 4.2

Multiply $\frac{3 a}{a + b}$ by $\frac{a - b}{a - b}$ .

$\frac{a (a + b)}{(a - b) (a + b)} + \frac{3 a (a - b)}{(a + b) (a - b)} - \frac{2 a b}{(a + b) (a - b)}$

Step 4.3

Reorder the factors of $(a - b) (a + b)$ .

$\frac{a (a + b)}{(a + b) (a - b)} + \frac{3 a (a - b)}{(a + b) (a - b)} - \frac{2 a b}{(a + b) (a - b)}$

Step 5

Simplify terms.

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Step 5.1

Combine the numerators over the common denominator.

$\frac{a (a + b) + 3 a (a - b)}{(a + b) (a - b)} - \frac{2 a b}{(a + b) (a - b)}$

Step 5.2

Combine the numerators over the common denominator.

$\frac{a (a + b) + 3 a (a - b) - 2 a b}{(a + b) (a - b)}$

Step 6

Simplify each term.

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Step 6.1

Apply the distributive property.

$\frac{a \cdot a + a b + 3 a (a - b) - 2 a b}{(a + b) (a - b)}$

Step 6.2

Multiply $a$ by $a$ .

$\frac{a^{2} + a b + 3 a (a - b) - 2 a b}{(a + b) (a - b)}$

Step 6.3

Apply the distributive property.

$\frac{a^{2} + a b + 3 a \cdot a + 3 a (- b) - 2 a b}{(a + b) (a - b)}$

Step 6.4

Multiply $a$ by $a$ by adding the exponents.

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Step 6.4.1

Move $a$ .

$\frac{a^{2} + a b + 3 (a \cdot a) + 3 a (- b) - 2 a b}{(a + b) (a - b)}$

Step 6.4.2

Multiply $a$ by $a$ .

$\frac{a^{2} + a b + 3 a^{2} + 3 a (- b) - 2 a b}{(a + b) (a - b)}$

Step 6.5

Rewrite using the commutative property of multiplication.

$\frac{a^{2} + a b + 3 a^{2} + 3 \cdot - 1 a b - 2 a b}{(a + b) (a - b)}$

Step 6.6

Multiply $3$ by $- 1$ .

$\frac{a^{2} + a b + 3 a^{2} - 3 a b - 2 a b}{(a + b) (a - b)}$

Step 7

Simplify terms.

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Step 7.1

Add $a^{2}$ and $3 a^{2}$ .

$\frac{4 a^{2} + a b - 3 a b - 2 a b}{(a + b) (a - b)}$

Step 7.2

Subtract $3 a b$ from $a b$ .

$\frac{4 a^{2} - 2 a b - 2 a b}{(a + b) (a - b)}$

Step 7.3

Subtract $2 a b$ from $- 2 a b$ .

$\frac{4 a^{2} - 4 a b}{(a + b) (a - b)}$

Step 7.4

Factor $4 a$ out of $4 a^{2} - 4 a b$ .

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Step 7.4.1

Factor $4 a$ out of $4 a^{2}$ .

$\frac{4 a (a) - 4 a b}{(a + b) (a - b)}$

Step 7.4.2

Factor $4 a$ out of $- 4 a b$ .

$\frac{4 a (a) + 4 a (- b)}{(a + b) (a - b)}$

Step 7.4.3

Factor $4 a$ out of $4 a (a) + 4 a (- b)$ .

$\frac{4 a (a - b)}{(a + b) (a - b)}$

Step 7.5

Cancel the common factor of $a - b$ .

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Step 7.5.1

Cancel the common factor.

$\frac{4 a (a - b)}{(a + b) (a - b)}$

Step 7.5.2

Rewrite the expression.

$\frac{4 a}{a + b}$